It comes after the US failed to agree visa-free travel for citizens of five EU countries – Bulgaria, Croatia, Cyprus, Poland and Romania – as part of a reciprocity agreement. US citizens can normally travel to all countries in the bloc without a visa.

The vote urges the revocation of the scheme within two months, meaning Americans will have to apply for extra documents for 12 months after the European Commission implements a “delegated act” to bring the change into effect.

The Commission discovered three years ago that the US was not meeting its obligations under the reciprocity agreement but has not yet taken any legal action. The latest vote, prepared by the civil liberties committee and approved by a plenary session of parliament, gives the Commission two months to act before MEPs can consider action in the European Court of Justice.

Australia, Brunei, Japan and Canada were also failing in their obligations, but all four have lifted, or are soon to lift, any visa restrictions on travel for EU citizens.

The Commission is legally obliged to act to suspend the visa waiver for Americans, but the European Parliament or the Council of the European Union have the chance to object to the “delegated act” it uses to do so.

In December, MEPs pressed for the move in order to “encourage” Washington to play its part, according to a statement by the parliament.

But Migration Commissioner Dimitris Avramopoulos warned of “consequences”, including potential “retaliation” and a drop in visitor numbers precipitating substantial losses for the continent’s tourism industry.

Just days ago the Council said it would liberalise the visa regime for citizens of Georgia travelling into the EU.

Georgians can now, subject to final approval of the regulation, stay in any EU country for 90 days in any period of 180 days without needing a visa.

Carmelo Abela, Malta’s minister for national security, said: “This agreement will bring the people of Georgia and the EU closer together and will strengthen tourism and business ties. It follows the completion of the necessary reforms by Georgia, addressing document security, border management, migration and asylum.”

Immigration minister Robert Goodwill told Parliament the EU was discussing the possibility of introducing a version of America’s Electronic System for Travel Authorisation (ESTA).

Currently foreign travellers must pay a fee of $14 (£11) when they complete ESTA, an automated online system that determines their eligibility to travel to the US.

“British people are now used to the US ESTA scheme and, therefore, we view with interest how the European scheme might develop and what similarities, and differences, there may be to the US scheme,” Mr Goodwill said.

“This type of scheme is generally there to help enhance security. To get to know as much as possible about the people who are intending to travel.

“It isn’t just flights, it could be people using ferries, or other border crossings into the European Union.”

Alan Brown, an SNP member of the European Scrutiny Committee, pointed out that Leave advocates in the referendum campaign had said there would be no need for visa-like travel schemes after Brexit.